Computing All Shortest Paths in a Network with Betweenness Centrality Analysis

We can compute all shortest paths in a network by implementing Dijkstra's algorithm, which efficiently finds the shortest paths between all node pairs through iterative relaxation of edge weights. The algorithm typically uses priority queues to select the next node with the minimum distance, ensuring optimal path discovery. From these identified shortest paths, we can determine which nodes and edges are most critical to network connectivity - those with higher betweenness centrality values. Betweenness centrality measures how often a node/edge appears on shortest paths between all node pairs, calculated by counting path traversals through each network element. Nodes and edges with higher betweenness values are more crucial for network connectivity and information flow, as they serve as important bridges or bottlenecks. This analysis can be applied to both directed and undirected graphs, requiring only minor adjustments in path traversal counting logic. The implementation typically involves: 1) shortest path matrix computation using Dijkstra's algorithm, 2) betweenness accumulation through path enumeration, and 3) normalization of centrality values. This process provides deep insights into network topology and flow dynamics, helping identify critical infrastructure elements in transportation systems, communication networks, or biological pathways.